Question

The differential coefficient of ${\sec }^{-1}\left(\frac{1}{2x^2-1}\right)$ w.r.t. $\sqrt{1-x^2}$ is-

• A. $\frac{1}{x^2}$
• B. $\frac{2}{x^3}$
• C. $\frac{x}{2}$
• D. $\frac{2}{x}$

Answer 1

The correct option is D $\frac{2}{x}$

$\frac{d}{dx}\left({\sec }^{-1}\left(\frac{1}{{2x}^2-1}\right)\right)=\frac{d}{dx}\left({\cos }^{-1}({2x}^2-1)\right)$
$=-\frac{4x}{\sqrt{1-{({2x}^2-1)}^2}}=\frac{-4x}{\sqrt{1-4x^2-1+{4x}^2}}$
$=-\frac{4x}{\sqrt{{4x}^2-{4x}^4}}=\frac{-2x}{\sqrt{x^2-x^4}}=-\frac{-2}{\sqrt{1-x^2}}$
$\frac{d}{dx}\left(\sqrt{1-x^2}\right)=\frac{-2x}{2\sqrt{1-x^2}}$
Therefore derivative of ${\sec }^{-1}\left(\frac{1}{{2x}^2-1}\right)$ w.r.t $\sqrt{1-x^2}$ is
$=\frac{-2}{\sqrt{1-x^2}}\times \frac{2\sqrt{1-x^2}}{-2x}=\frac{2}{x}$